Input and Output Manifold Constrained Gaussian Process Regression for Galvanometric Setup Calibration

Abstract

Data-driven techniques are finding their way into the calibration procedure of galvanometric setups. However, they bypass the underlying physical or mathematical model completely. Recent work has shown that a simple assumption about an underlying truth can improve the predictions: laser beams leaving the device follow a straight line. In this article, we take that approach a step further. Both the inputs (the pairs of rotations of the two mirrors) and outputs (the straight lines) lie on a manifold. We can incorporate this prior knowledge in the model via constraints built in the formulation of a covariance function. We propose two constrained models: one in which a linear constraint on the direction vector is written as a differential equation, and one in which a quadratic constraint is imposed by a reparametrization of the line coordinates. We compare them with the data-driven and unconstrained model-based approaches. We show that enforcing constraints improves the quality of the predictions significantly and, thus, the accuracy of the calibration. We validate our findings against real world data by predicting points on validation planes, calculating line segment distances, considering the training times for the models, and assessing how much a predicted line resembles an actual straight line.